The Annals of Probability

Poly-logarithmic localization for random walks among random obstacles

Jian Ding and Changji Xu

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Abstract

Place an obstacle with probability $1-\mathsf{p}$ independently at each vertex of $\mathbb{Z}^{d}$, and run a simple random walk until hitting one of the obstacles. For $d\geq2$ and $\mathsf{p}$ strictly above the critical threshold for site percolation, we condition on the environment where the origin is contained in an infinite connected component free of obstacles, and we show that the following path localization holds for environments with probability tending to 1 as $n\to\infty$: conditioned on survival up to time $n$ we have that ever since $o(n)$ steps the simple random walk is localized in a region of volume poly-logarithmic in $n$ with probability tending to 1. The previous best result of this type went back to Sznitman (1996) on Brownian motion among Poisson obstacles, where a localization (only for the end point) in a region of volume $t^{o(1)}$ was derived conditioned on the survival of Brownian motion up to time $t$.

Article information

Source
Ann. Probab., Volume 47, Number 4 (2019), 2011-2048.

Dates
Received: September 2017
Revised: July 2018
First available in Project Euclid: 4 July 2019

Permanent link to this document
https://projecteuclid.org/euclid.aop/1562205700

Digital Object Identifier
doi:10.1214/18-AOP1300

Mathematical Reviews number (MathSciNet)
MR3980914

Subjects
Primary: 60K37: Processes in random environments 60H25: Random operators and equations [See also 47B80] 60G70: Extreme value theory; extremal processes

Keywords
Random walk among random obstacles localization

Citation

Ding, Jian; Xu, Changji. Poly-logarithmic localization for random walks among random obstacles. Ann. Probab. 47 (2019), no. 4, 2011--2048. doi:10.1214/18-AOP1300. https://projecteuclid.org/euclid.aop/1562205700


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